# Compton scattering

As Compton effect is defined as the increase in the wavelength of a photon in the scattering of a particle. First the Compton effect was observed of electrons. This Compton scattering (after Arthur Compton ) is an important ionization process and the dominant interaction process of high-energy radiation with matter for photon energies between 100 keV and 10 MeV.

- 7.1 Inelastic scattering by elastic collision
- 7.2 (In) elastic scattering and internal degrees of freedom

- 8.1 Resting electron
- 8.2 Any reference system

## History

Until the discovery of the Compton effect, the photoelectric effect was the only finding that light not only as a wave, but also, as postulated from Albert Einstein in 1905 as a stream of particles behaves (see wave -particle duality ).

When Arthur Compton investigated the scattering of high-energy X-rays of graphite in 1922, he made two observations: First, the scattering angle distribution was in forward and reverse direction is not the same and the other, the wavelength of the scattered radiation was greater than that of the incident radiation. Both observations were incompatible with the idea of an electromagnetic wave scattered by free electrons will (Thomson scattering) or of bound electrons ( Rayleigh scattering). The electrons would oscillate with the frequency of the incident wave and emit a wave with unchanged frequency.

Instead Compton measurements showed that the wavelength of the scattered radiation depending on the scattering angle as in a collision of particles, the photon and the electron, behaves ( derivation see below). This Compton proved the particle nature of light - or the wave nature of electrons, because electrons are treated as matter waves and light as an electromagnetic wave, it follows as in the above Feynman turn of the Compton effect.

## Compton wavelength

When shock of a (quasi ) free electron at rest this takes some of the energy of a photon whose energy is reduced to - it is an elastic collision. The larger its initial energy, the more complete the energy can be transferred, right shown. In a " glancing blow " with the photon retains almost all its energy, with a " head-on collision " with the photon is scattered back and lose a maximum of energy. Here, the angle by which changes the moving direction of the photon.

Due to the energy loss increases the wavelength of the photon. It is noteworthy that this increase depends only on the angle, and not from the original photon energy:

The Compton wavelength is a characteristic size of a particle with mass. It indicates the increase in the wavelength of the scattered at right angles to him photon.

Is the Compton wavelength of a particle of mass

Where is Planck's constant and the vacuum speed of light.

Often (especially in elementary particle physics ) and the reduced Compton wavelength is used with the reduced Planck 's constant and even without the addition referred to as reduced Compton wavelength. In this form, the Compton wavelength emerges as a parameter in the Klein-Gordon equation.

### Compton wavelength of electron, proton and neutron

The Compton wavelength of electrons, protons, neutrons and are therefore, unlike the de Broglie wavelength of its speed independently; their values are according to the current measurement accuracy:

The bracketed numbers indicate the estimated standard deviation of the mean corresponding to the last two digits before the brackets.

The reduced Compton wavelength of the electron is approximately 400 femtometer or more precisely, the proton and neutron round or 0.2 femtometer.

The very small changes in wavelength are the reason that the Compton - effect can be observed only at very short wavelength radiation, in the area of X-ray and gamma radiation. At long wavelength the relative increase is small, the scattering seems to occur without loss of energy, this is called Thomson scattering.

## Scattering cross-section

The angle-dependent cross section for Compton scattering (in the approximation of free, static electrons ) is given by the Klein-Nishina formula. In the Compton scattering in matter an electron from the atomic shell is struck. In this case, these formulas are only approximate. The influence of the momentum of the electron bound to the energy of the scattered photon is called Doppler broadening. It is the projection of the pulse distribution of the scattered electrons to the direction of the momentum transfer during the scattering. It is particularly pronounced at low energy, large scattering angles and atoms with high atomic number.

Interspersed one photon to other objects than electrons, for example on a proton, the mass must be adjusted accordingly in the above equations, thereby Compton wavelength and cross section would change.

## Inverse Compton effect

The inverse Compton effect scatters a high energy electron (or other charged particle, such as a proton ) on a low-energy photon and transferring energy to the photon. The inverse Compton effect occurs in particle accelerators and can be observed in astrophysical outflows in the coronae of accretion disks of active galactic nuclei and supernovae (see also Sunjajew - Seldowitsch effect). Inverse Compton scattering on the background radiation limits the maximum energy of protons in cosmic rays (see also GZK cutoff ).

## Applications

Since it is very difficult to focus the gamma radiation by means of lenses, the Compton effect plays an important role in imaging by means of gamma rays in the energy range of a few hundred kilo-electron volts to several tens of mega electron volts. In so-called Compton telescopes (also known as Compton cameras) do you measure energy and direction of the scattered photon, and energy and (sometimes) even the direction of the electron. Thus, energy, direction of origin and possibly the polarization of the incident photon can be determined. In reality, this is greatly complicated by uncertainties and unmeasured variables such as the direction of the electron, so that complex event and image reconstruction methods must be used.

The most well-known Compton telescope COMPTEL was that of 1991 as a first telescope explored the starry sky in the energy range from 0.75 to 30 MeV on board the NASA satellite CGRO to 2000. Among the successes of COMPTEL include, inter alia, the creation of the first maps of the sky in this energy range, the study of nucleosynthesis, for example, of radioactive 26Al ( massive stars and supernovae ) and 44Ti and progress in the study of pulsars, active galaxies ( AGNs ) etc.

It run development work on the use of Compton cameras in the field of medicine or nuclear technology. In medicine, they may have had against the scintigraphic gamma cameras used today produce images with better spatial resolution, ie locate tumors and metastases more accurately. In nuclear engineering, for example, nuclear power plants or nuclear waste could be monitored in the future by means of Compton cameras.

For the security checks at airports scanner devices have been developed which utilize the Compton backscattering (English backscatter ) of X-rays on surfaces. These are currently being tested in the United States.

The inverse Compton effect is used to generate by backscattering of laser photons by energetic electrons highly monochromatic, linearly polarized gamma radiation.

## Compton continuum and Compton edge

From the formulas derived below are calculated easily an expression for the angle-dependent energy of the photon and the kinetic energy of the electron after scattering ( Klein-Nishina formula ):

- Photon:
- Electron:

If many photons of energy after Compton scattering (eg in a scintillator detector or other ), so there is a characteristic energy spectrum of the scattered electrons, as shown in the adjacent graph. Which in this case transferred to the electron energy is a continuous function of the scattering angle ( Compton continuum ), but has a sharp upper limit. This so-called Compton edge arises because the scattered photons transfer the maximum energy to the electrons at = 180 °. Thus, the edge in the spectrum is

In addition, we obtain the energy spectrum of a " photopeak " or " Full Energy Peak", a spectral line at the energy. It comes from detection events, in which the total energy of the photon is deposited in the detector, for example, by the photoelectric effect. From the above formula can be seen that that pertains to the photopeak to Compton edge

The left of this peak is located.

The figure at right shows a with a germanium detector (see gamma spectroscopy) spectrum recorded. At about 4.4 MeV we find the photopeak of the gamma radiation that comes from inelastic neutron scattering on 12C nuclei (the line is dopplerverbreitert by recoil movement of the carbon nuclei). From the gamma energy 4.4 MeV follows from the equation above that the corresponding Compton edge will be about 4.2 MeV, where it is also easy to recognize in the figure. To her left shows the corresponding continuum with seated thereon peaks of other origin.

## Elastic or inelastic scattering

The Compton effect is not uniformly identified in the literature as inelastic scattering or elastic scattering. Right or wrong does not exist in this context, but it depends on the way of looking at.

Properly, however, is that it is an elastic collision with the Compton scattering, because this is the one prerequisite for the calculation (energy and momentum conservation ), on the other hand all the properties an elastic collision are met.

### Inelastic scattering by elastic collision

Although it is equated in some books, unfortunately, often, an elastic collision must not automatically mean that it is elastic scattering. At scattering processes, in particular in the scattering of photons only the scattered object is often considered. In the Compton effect, the photon loses by interaction with the scattering object energy (it changes frequency / wavelength), so some physicists speak here of inelastic scattering.

### (In) elastic scattering and internal degrees of freedom

In the literature inelastic processes but often also be explained that internal degrees of freedom are excited. A single free electron has no internal degrees of freedom but according to our present understanding. One might say that internal degrees of freedom of the atom are excited only when an electron weakly bound to an atom. From this perspective, one can speak of elastic scattering.

In summary it can be stated, therefore, that it is no longer possible in connection with the Compton effect, define a consistent use of the concept of inelastic or elastic scattering. One should instead be aware that the Compton effect, the photon interaction by some of its energy to deliver the scattering object ( electron, proton, ...). Whether the viewer is for the description of the Coating now sees the release of energy as much and therefore speaks of inelastic scattering, or whether he wants to instead rely on the definition of the internal degrees of freedom, depends in most cases on the context or the other party. Depending on can be considered both as an elastic and inelastic scattering as Compton scattering so.

## Derivation of the Compton formula

The different derivations a free electron is always assumed. If the electron is bound in an atom, you have to remove the binding energy of the kinetic energy of the electron after the collision.

### Electron at rest

In the following, we calculate the Compton formula by assuming the particles than at the beginning dormant. In the scattering, the photon transfers part of its energy to the electron, such that the two particles move in different directions by the scattering apart.

First we look at the energy and momentum which the respective particles before and after scattering wear ( stands for the frequency and for the scattering angle):

The two particles must satisfy the energy and momentum conservation law before and after the scattering.

Since we still need an expression for, we make the conservation of energy to it and set the appropriate sizes a.

In the special theory of relativity are the energy and momentum of a particle about the so-called energy - momentum relation interrelated. Furthermore, we need the law of cosines from trigonometry, since the particles move to the sides of a triangle, which correspond to their respective pulse. We use him for us to express the unknown momentum of the electron by the momentum of the photon.

Now we use our expressions for and into the energy - momentum relation, multiply all the staples out and summarize the many terms together

In the last step we have used that the wavelength and frequency are related to each other via ( wave equation ).

### Any reference system

As can be calculated in the case of the electron at rest slightly trigonometrisch the Compton effect, the situation is more complex when we look at them from any frame of reference. In this case, the electron moves before the collision with the speed, carrying with it the total energy and momentum,

With and.

To calculate the Compton effect in the case now under consideration, we use the four-vector formalism.

The four-momenta which have the particles involved before and after the scattering process are

Herein, a unit vector pointing in the direction of the photon.

Since we will need the products of two of the individual pulses, we now want to calculate this. For the squares always

It being understood that the following applies, as it is in a vector of length 1.

Now we still need the mixed products:

When is the angle between the electron and the photon before scattering.

Are obtained analogously

Wherein the angle between the electron and the photon before the dispersion after the dispersion is.

Wherein the angle between the photon before the photon scattering and after scattering is.

A photon scattered by an electron, the energy and momentum must be met. Since energy is proportional to the zero sequence component of the four- pulse, and the remaining components represent the impulse follows

Using and this simplifies to

After inserting the previously computed components, we obtain

Depending on the angle of incidence and kinetic energy the electron may be some energy transferred to the photon ( inverse Compton scattering). In the rest frame of the electron, the speed was the same zero before the collision. It is always advisable

Bringing the already known formula

Results.